US7991240B2 - Methods, systems and apparatuses for modeling optical images - Google Patents

Abstract

Methods, systems and apparatuses for modeling high resolution images which significantly shortens computation time and reduces image artifacts, as compared to known methods. Embodiments implement a look-up table, to compute once and store the point spread functions for various points of an image. During modeling of the optical system, previously stored point spread functions may be used to determine the point spread function for a specified point using a weighted interpolation of the point spread functions that have been stored for nearby points.

Description

FIELD OF THE INVENTION

Disclosed embodiments relate generally to modeling systems and more particularly to methods, systems and apparatuses for modeling an optical system.

BACKGROUND OF THE INVENTION

When designing optical systems, such as cameras, sensors, optical instruments, electro-optical devices, etc., designers use computational modeling to test the performance of the designs. By modeling the scene image output of optical systems, designers can predict errors before the optical system is made or deployed. Therefore, these errors can be corrected before the optical system is actually built. Thus, modeling the scene image output saves both time and resources that could have been wasted by building a design that does not meet the desired specification.

FIG. 1 illustrates a typicaloptical system1. Aslight rays30 from an object scene pass through alens10, they intercept an image plane40, thereby forming an image on the plane40. InFIG. 1, the image plane40 happens to be located on animage sensor20, so the image formed on the image plane40 corresponds to the image received atimage sensor20.

FIG. 2aillustrates animage50 formed at the image plane40 of the optical system1 (FIG. 1). In order to model this image, designers view the image as having many points, such as point P. Through computation, designers are able to determine the image characteristics of theimage50 at the various points P throughout theimage50. Image characteristics might vary depending on qualities associated with thelens10, such as the lens prescription.

Designers are able to specify the location of points P in the image using a polar coordinate system, such as that illustrated inFIG. 2b. Specifically, in a polar coordinate system, each point can be described by a radial distance r and an angle θ. Distance rp represents the length between the origin O of the polar coordinate system and the selected point P in theimage50. For example,FIG. 2bshows a point P that is rp length units away from the origin O. The length rp may be represented using nanometers or any other unit of length. Angle θ represents the angle between the selected point P and the polar axis x. For example, point P inFIG. 2bis θp angular units away from the polar axis (e.g., 0°). The angle θ may be represented using degrees, radians, or any other unit for describing an angle.

In addition to modeling characteristics at a point P in theimage50 according to the point's physical location (rp, θp), designers may also model characteristics specific to different wavelengths of light at point P. Thus, wavelength λp is also associated with point P.

One particular image characteristic that designers would like to have information about before building an optical system is the impulse response at various points and wavelengths in a simulated image. This impulse response is also known as the point spread function. The point spread function of an optical system is a widely used characteristic that describes the response of the optical system to a point source or a point object and also describes the amount of blur introduced into a recorded image. Traditionally, optical ray tracing programs, such as ZEMAX®, available from ZEMAX® Development Corp., 3001 112th Ave. NE, Bellevue, Wash. 98004, have been used to compute this data for each point in the image. However, modeling using ray tracing to determine the point spread function for each point in the image is impractical because of its high computational demands, especially as image sizes and resolutions have increased. Known methods also result in undesirable image artifacts in the simulated image.

Accordingly, there exists a need in the art for improved techniques for determining the point spread function at points in a simulated image from an optical system.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an optical system.

FIG. 2aillustrates an image scene formed on an image plane.

FIG. 2billustrates an image scene formed on an image plane, illustrating a polar coordinate system.

FIG. 3 illustrates a data processing system in accordance with disclosed embodiments.

FIG. 4 illustrates a method of determining a point spread function at a point on an image scene in accordance with disclosed embodiments.

FIGS. 5a-5billustrate a look-up table in accordance with disclosed embodiments.

FIGS. 6a-6cfurther illustrate the look-up table ofFIGS. 5a-5bin accordance with disclosed embodiments.

FIG. 7 is a flowchart illustrating an interpolation process in accordance with disclosed embodiments.

FIG. 8 is a diagram illustrating points in the look-up table and an input point in accordance with disclosed embodiments.

FIG. 9 is a flowchart illustrating another interpolation process in accordance with disclosed embodiments.

FIG. 10 is another diagram illustrating points in the look-up table and an input point in accordance with disclosed embodiments.

FIG. 11 is a flowchart illustrating a modeling process in accordance with disclosed embodiments.

DETAILED DESCRIPTION

In the following detailed description, reference is made to the accompanying drawings which form a part hereof, and in which are shown, by way of illustration, specific embodiments. These embodiments are described in sufficient detail to enable those skilled in the art to make and use them, and it is to be understood that structural, logical or procedural changes may be made. Particularly, in the description below, processes are described by way of flowchart. In some instances, steps which follow other steps may be reversed, be in a different sequence or be in parallel, except where a following procedural step requires the presence of a prior procedural step.

Disclosed embodiments may be implemented using any type of data processing system.FIG. 3 illustrates one example of such adata processing system60. Specifically,FIG. 3 shows the basic components of acomputer system60, including a central processing unit (CPU)70, which executes software instructions, random access memory (RAM)80, an input/output (I/O)device100,removable memory110, and adata bus90 that allows the other parts of thesystem60 to communicate. As is known in the art,computer system60 can be operated according to software instructions stored in any of its various memories. Methods of programming thesystem60 to implement the disclosed embodiments are also known in the art. Additionally, while one could use thesystem60 to implement the various disclosed embodiments, one could also use any other type of data processing system.

Disclosed embodiments provide methods, systems and apparatuses for modeling high resolution images that significantly shortens computation time and reduces image artifacts, as compared to known methods. Embodiments implement a look-up table, to compute once and store the point spread functions for various combinations of location (r, θ) and wavelength (λ), representing various points of theimage50. During modeling of the optical system, these previously stored point spread functions may be used to determine the point spread function for a specified point using a weighted interpolation of the point spread functions that have been stored for nearby points. The embodiments do not require the computation heavy ray tracing technique for each point in the image. This significantly reduces the computation time required for optical system modeling.

Referring now toFIG. 4, one disclosed embodiment is now described in more detail.FIG. 4 is a flowchart illustrating anoverall method135 performed bydata processing system60. First, atstep140, themethod135 computes and stores a look-up table (“LUT”) that associates point spread functions (“PSF”s) with various points, described as combinations of location (r, θ) and wavelength (λ). As previously described, a point spread function describes the response of the optical system to a point source or a point object as well as the amount of blur introduced into a recorded image. As described in more detail below, the look-up table represents a model of animage50 formed on an image plane40 inoptical system1. The look-up table could be generated with the samedata processing system60 used to perform the other steps shown inFIG. 4 or with a different data processing system. The look-up table may contain, for example, 360 stored point spread functions for each wavelength as opposed to the indefinite number of point spread functions which may need to be modeled for a given wavelength.

Atstep150, the location (in polar coordinates) of a point P inimage50 and a wavelength of light for a point for which modeling is desired is input intosystem60. Atstep160,method135 determines whether the look-up table contains a stored location and wavelength that exactly matches the input location and wavelength. If the look-up table includes the exact combination of location (r, θ) and wavelength (λ), thenmethod135 outputs the point spread function associated with the location and wavelength atstep170. Ifmethod135 determines that the input location and wavelength do not exactly match values stored in the look-up table, thenmethod135 performs interpolation operations atstep180. The point spread function based on the interpolation is output atstep190. As will be explained in more detail below, the interpolation operations determine a point spread function for the input point's location and wavelength, based on the point spread functions stored in the look-up table that are associated with points near the input point's location and/or wavelength. Each of the steps illustrated inFIG. 4 is described in greater detail below.

With reference toFIGS. 5aand5b, the determination of the look-up table values (step140) is now described in more detail.FIG. 5aillustrates animage50 to which a portion of a look-up table corresponds. For a given wavelength, in this case λ0, theimage50 is divided into a set of radial bands, r0 to rn, duringstep140. Each radial band has a “radial height,” which is the band's distance from a point inimage50 designated as the origin O. In disclosed embodiments, the distance between the radial bands is the same; however embodiments of the invention are not limited as such. Also duringstep140, theimage50 may also be divided into a set of angular positions.FIG. 5ashows image50 divided into seven radial bands (r0 to r6) and 36 angular positions (0°, 10°, . . . , 340°, 350°) Note, only the angular positions from 0° to 90° are shown inFIG. 5a. It should also be noted that the numbers of radial bands and angular positions are not limited to those shown inFIG. 5a. For example, the look-up table could include 360 angular positions (every 1°). Larger look-up tables will provide a higher accuracy, but as a trade off, they require more processing time to compute and to utilize during subsequent modeling steps.

The look-up table based on theimage50 shown inFIG. 5aincludes stored point spread functions associated with the intersection of each angular position and radial band (shown as ● inFIG. 5a). Methods of determining these point spread function values are well known; one way the determination can be done is using ray tracing programs, such as ZEMAX®, available from ZEMAX® Development Corp., 3001 112th Ave. NE, Bellevue, Wash. 98004. The point spread functions are based on characteristics of the optical system, such as the lens prescription. For example, by modeling the optical system shown inFIG. 1 using ray tracing systems, designers can determine the point spread functions ofimage50 at radial bands r0 to r6 for angular positions of 0°, 10°, 20°, etc. In other embodiments, designers are able to generate a look-up table associated with any image plane inoptical system1, such as image planes located betweenlens10 andimage sensor20, using the same such ray tracing programs, and are not limited to images formed at image plane40.

FIG. 5billustrates a set ofimages50 to which a larger portion of the look-up table corresponds. As can be seen fromFIG. 5b, point spread functions can also be determined based on various wavelengths in addition to various radial bands and angular positions. For example, using known methods, such as ray tracing programs like ZEMAX® according to lens prescriptions, one can determine a set of point spread functions for the first wavelength λ0 as well as for other wavelengths, such as λ1 and λ2 shown inFIG. 5b. Individual point spread functions are each associated with a specific combination of wavelength, radial band, and angular position.

In one disclosed embodiment, point spread functions are determined for a variety of wavelengths that are within the visible spectrum of light. For example, the spectrum of light visible to the human eye can be divided into bands, then point spread functions are determined for each wavelength in the middle of each band. In such an embodiment, for the spectrum of light between 400 nm to 700 nm, one could generate point spread functions in 10 nm intervals (400 nm, 410 nm, 420 nm, etc. up to 700 nm). In a similar manner where the number of radial bands and angular positions may be varied to accommodate the simulation requirements, the wavelength interval could also be varied. Additionally, if point spread functions values are generated for only every 50 nm of wavelength instead of every 10 nm, a look-up table requiring even less memory storage space could be generated.

FIG. 6ais a table representing a portion of a look-up table for a single wavelength (λ0). The table ofFIG. 6acorresponds to a diagram similar to that ofFIG. 5a, but includes data representing only three radial bands (r0 to r2) and 360 angular positions (0° to 359°). The rightmost column includes each point spread function associated with the corresponding combination of wavelength λ, radial band r, and angular position θ shown in the adjacent columns of the same row.

FIGS. 6band6crepresent additional portions of the look-up table (similar toFIG. 6a).FIG. 6bincludes point spread functions for wavelength λ1.FIG. 6cincludes point spread functions for wavelength λ2. TogetherFIGS. 6a,6band6crepresent, in tabular form, a diagram similar to that ofFIG. 5b, exceptFIGS. 6a,6band6conly include data for three radial bands and include data for 360 angular positions.

Referring back toFIGS. 3 and 4, once the look-up table is computed and stored in system60 (step140), it can then be used to determine the point spread function for any point and wavelength on theimage50 formed at image plane40 (FIG. 1). Atstep150 the polar coordinate and wavelength of a point Pn is input into themethod135. If the polar coordinate and wavelength of the point Pn perfectly matches a polar coordinate and wavelength already stored in the look-up table, thes method135 can simply output the point spread function associated with the input polar coordinate and wavelength (step170). However, if the polar coordinate and wavelength of Pn is not stored in the look-up table, then atstep180 themethod135 performs an interpolation process, which is described below.

Referring toFIGS. 7 and 8, the interpolation process (step180) is now described in more detail.FIG. 7 is a flowchart illustrating the process ofstep180.FIG. 8 is a two-dimensional diagram illustrating the point spread functions that have been pre-calculated (step140) for the specified wavelength λ1 that correspond to points near the point that was input atstep150. Specifically,FIG. 8 shows the coordinate input at step150: (rp, θp). Additionally,FIG. 8 shows the four surrounding positions: coordinate (r1, θ1), coordinate (r1, θ2), coordinate (r2, θ1) and coordinate (r2, θ2). As explained above, the point spread functions at the four coordinates surrounding the input point (rp, θp), were previously generated and stored in system60 (step140). Note that the point spread function at coordinate (rp, θp) is unknown because it was not computed during the look-up table generation process.

Atstep200, the values of the four points in the look-up table surrounding the input point are determined. The values of r1 and θ1 are determined according to Equations (1) and (2):
r1=floor(rp), and  (1)
θ1=floor(θp).  (2)
The “floor” function returns the largest whole values of r and θ below rp and θp, respectively. Next, r2 is determined by adding to r1 the distance between the radial height of r1 and the radial height of r2. Then, θ2 is determined by adding the predetermined angular interval to θ1. For example, if the look-up table was generated having 36 angular position, r2 and θ2 would be calculated according to Equations (3) and (4):
r2=r1+1, and  (3)
θ2=θ1+10°.  (4)
Similarly, if the look-up table was generated with 360 angular positions (as opposed to 36), then only 1° would be added to θ1 in Equation (4). The four look-up table points surrounding the input point are (r1, θ1), (r1, θ2), (r2, θ1) and (r2, θ2).

After determining the four look-up table points surrounding the input point, the point spread functions associated with the four points atstep210 are extracted. As shown inFIG. 8, these values correspond to PSF(r1, θ1), PSF(r1, θ2), PSF(r2, θ1) and PSF(r2, θ2).

Once the four surrounding point spread functions have been determined, the point spread function associated with the input point through weighted linear interpolation can be determined atsteps220 and230. This interpolation involves determining the distances between the input point and the surrounding points to generate weighting values, then multiplying the pre-calculated point spread functions by the weighting value.

Angular interpolation is performed atstep220. An angular interpolation procedure is shown in Equations (5), (6) and (7):
wt_angle=θ_p−θ₁  (5)
PSF(r1,θp)=(1−wt_angle)*PSF(r1,θ1)+(wt_angle)*PSF(r1,θ2)  (6)
PSF(r2,θp)=(1−wt_angle)*PSF(r2,θ1)+(wt_angle)*PSF(r2,θ2)  (7)
where wt_angle is the weighting value in the angular direction and is equal to the angular distance between point P andpoint1 and PSF(r1, θp) and PSF(r2, θp) are the point spread functions for points located atradial locations1 and2, but with the angular location of point P.

Next, atstep230, radial interpolation is performed to determine the PSF at point (rp, θp). The following equations illustrate this procedure:
wt_radial=rp−r1;  (8)
PSF(rp,θp)=(1−wt_radial)*PSF(r1,θp)+(wt_radial)*PSF(r2,θp);  (9)
where, similar to above, wt_radial is the weighting value in the radial direction and is equal to the radial distance between point P andpoint1 and PSF(rp, θp) is the point spread function for point P. Once PSF(rp, θp) is determined, the function that has been generated is output atstep240.

FIG. 7 shows a process of determining the point spread function at a single point in an image plane using a pre-calculated look-up table. Of course, designers will often need to know the point spread functions at points throughoutimage50. This may be accomplished by repeating the steps shown inFIG. 7 for various points P located throughoutimage50.

The process illustrated byFIGS. 7 and 8 assumes the input point is associated with a wavelength included in the look-up table, such as λ1, λ2, or λ3 shown inFIGS. 5a-6c. However, thesystem60 andmethod135 can also be used to determine the point spread function for wavelengths that have not been pre-calculated and stored in the look-up table, using a similar interpolation process.FIG. 9 illustrates a flowchart of such a process.FIG. 10 is a diagram showing the input value in relation to pre-calculated values for first and second wavelengths λ1 and λ2.

Atstep260, the point spread function at point (rp, θp) is determined for the first wavelength, λ1. Atstep270, the point spread function at point (rp, θp) is determined for the second wavelength, λ2.Steps260 and270 are implemented in accordance with the process shown inFIG. 7. Once these two point spread functions have been determined, a wavelength interpolation process that is similar to the angular and radial interpolation processes described above can be performed atstep280. First, a weighting value based on the distance between the input wavelength (λp) and the next lower wavelength for which point spread functions are stored in the look-up table (λ1) is determined. Based on this weighting value, the point spread function at (λ1, rp, θp) can be determined by interpolation, in accordance with Equations (10) and (11):
wt_—w1=λp−λ1;  (10)
PSF(λp,rp,θp)=(1−wt_—w1)*PSF(λ1,rp,θp)+(wt_—w1)*PSF(λ2,rp,θp);  (11)
where, similar to above, wt_w1 is the weighting value in the wavelength direction and is equal to the distance between the wavelengths of point P andpoint1 and PSF(λp, rp, θp) is the point spread function for point P. Atstep290, the determined point spread function for the input point is output.

The point spread function data determined in accordance with disclosed embodiments may be used to perform a convolution operation of an ideal image at each point position, as is known in the art, to complete the modeling of the optical system.

The disclosed embodiments have been described herein using polar coordinates. However, it should be noted that embodiments could use any other type of coordinate system, such as a Cartesian coordinate system.

As a further benefit, the size of the look-up table may be reduced. To reduce the size of the look-up table, its values can be downsampled. Downsampling means that the samples rate of a signal is reduced. This is usually done to reduce the data rate or the size of the data. For example, the resolution of the look-up table could be downsampled to match the resolution of theimage sensor20. Additionally, the point spread function data can be resampled to match the resolution of the optical image.

The above described methods of determining the point spread function of a point on an image plane can also be used as part of a larger modeling operation that includes, for example, the use of a distorted image grid and/or relative illumination factors.FIG. 11 illustrates one such operation. After establishing the look-up table atstep300, a grid of distorted image (“DI”) plane coordinates that correspond to the field height coordinates used in the look-up table can be determined (step310). Next, atstep320, relative illumination factors (“RI”) on the image plane corresponding to the field height coordinates from a raytracing program or from experimental measurements can be determined. The input scene could then be scaled according to the DI grid (step340) and the amplitudes of each point of the input scene could then be scaled according to the RI data (step360).Steps310,320,340 and360 are performed in accordance with known methods in the art. The point spread functions at each point of the input scene could be determined (step380) using the processes described above and applied to the input scene. Then the modeled scene could be output as a bitmap image (step390).

While various embodiments have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art(s) that various changes in form and detail can be made therein.

Claims (9)

1. A method of determining a point spread function at a specified point in an optical image, the method comprising:

with computing equipment, generating a data structure associating predetermined point spread functions with predetermined points in the optical image and with wavelengths;

with the computing equipment, determining the point spread function at the specified point according to the predetermined point spread functions, wherein the point spread function associated with the specified point is not stored in the data structure; and

with the computing equipment, determining the point spread function based on a specified wavelength by interpolating a difference between the specified wavelength and a wavelength associated with at least one of the predetermined point spread functions, wherein the point spread function associated with the specified wavelength is not stored in the data structure.

2. The method ofclaim 1, further comprising:

with the computing equipment, determining the point spread function at the specified point by interpolation of a plurality of the predetermined point spread functions.

3. The method ofclaim 2, wherein the interpolation is based on a plurality of the predetermined point spread functions associated with points surrounding the specified point.

4. The method ofclaim 2, wherein the interpolation is based on distances between the specified point and a plurality of points surrounding the specified point.

5. The method ofclaim 4, wherein the distances include a radial distance component and an angular component.

6. The method ofclaim 1, further comprising:

with the computing equipment, scaling the optical image in accordance with a predetermined grid of distorted image plane coordinates.

7. The method ofclaim 1, further comprising:

with the computing equipment, scaling the optical image in accordance with predetermined relative illumination factors.

8. A non-transitory computer-readable storage medium comprising:

instructions causing a data processing system to implement a method comprising:

generating a data structure storing point spread functions associated with points in an optical image and with wavelengths;

determining the point spread function at the specified point in the optical image based on the stored point spread functions, wherein the point spread function associated with the specified point is not stored in the data structure; and

determining the point spread function based on a specified wavelength by interpolating a difference between the specified wavelength and a wavelength associated with at least one of the predetermined point spread functions, wherein the point spread function associated with the specified wavelength is not stored in the data structure.

9. The non-transitory computer-readable storage medium ofclaim 8, wherein the instructions cause the data processing system to implement a method further comprising determining the point spread function at the specified point by interpolation of a plurality of the predetermined point spread functions, wherein the plurality of the predetermined point spread functions are associated with points surrounding the specified point.

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